Games and the Impossibility of Realizable Ideal Functionality

A cryptographic primitive or a security mechanism can be specified in a variety of ways, such as a condition involving a game against an attacker, construction of an ideal functionality, or a list of properties that must hold in the face of attack. While game conditions are widely used, an ideal functionality is appealing because a mechanism that is indistinguishable from an ideal functionality is therefore guaranteed secure in any larger system that uses it. We relate ideal functionalities to games by defining the set of ideal functionalities associated with a game condition and show that under this definition, which reflects accepted use and known examples, bit commitment, a form of group signatures, and some other cryptographic concepts do not have any realizable ideal functionality.